Gausian curvature

Gaussian curvature is a measure of how a surface bends at a specific point, considering all directions simultaneously. It combines the curvatures in two perpendicular directions—like along a hill's slope and across it—to determine the surface's overall shape at that point. If the surface curves the same way in every direction (like a sphere), the Gaussian curvature is positive. If it curves differently in different directions (like a saddle), it's negative. When the surface is flat at that point (like a table), the Gaussian curvature is zero. This concept helps us understand the intrinsic shape of surfaces in geometry and in nature.